Variability of total claim amounts under dependence between claims severity and number of events

Dependencies among random numbers of summands in random sums or among summands have been widely studied in recent literature, and several applications have been developed in actuarial sciences and reliability, where dependencies among risks or lifetimes are quite usual in common problems. In this note we derive stochastic comparisons for random sums when the summands and number of summands depend on a common random environment. The main results presented here describe how such dependence among the random environmental parameters influences the variability of random vectors of sums. As a corollary, we provide stochastic bounds for total claim amounts. Other applications and a discussion of a related result are included.